You are not missing anything, you are exactly correct!

The confusion comes about from terminology and an omission of rigor that would be distracting for most purposes. When the articles you mentioned, and many like them, say "focal length measures 'the distance, in millimetres, between the optical centre of the lens and the camera’s sensor (or film plane),'" they leave out the implied subject at infinity portion.

Realistically the difference between subjects at infinity and pretty damn close is insignificant when you consider the sensor plane / subject distance ratios as applied to the lens law equation you quoted.

The terminology becomes fractious when you are talking about the imaging distance of a subject at close range. When the image is in-focus is that the focal length? It's the length required for focus but the definition of focal length is for subject at infinity.

It's really just word games, the lens law math is reality.

Fortunately for photographers, assuming a constant effective focal length equal to the defined infinite distance definition approximates well within necessary parameters through a very large range of practical use.

This difference becomes more significant with very close or Macro Photography. To quote from Effective Aperture and Macro

Put simply, when working at greater magnifications, roughly 1:2 or more, the displayed aperture on your lens or in your camera will be slightly different from what the true f-stop is, and this number will continue to change as the magnification of your shot increases. This is due to the focal length of the lens beginning to change as focus extension changes; since the lens is physically further away from the sensor or film, there is a change in exposure and f-stop.